Solve frac{8^{8}u^{7}v^9}{8^10u^7v^3}

$$\frac { 8 ^ { 8 } u ^ { 7 } v ^ { 9 } } { 8 ^ { 10 } u ^ { 7 } v ^ { 3 } }$$

Evaluate

$$\frac{v^{6}}{64}$$

Steps Using Quotient of Powers Property

Use the rules of exponents to simplify the expression.

$$\frac{8^{8}u^{7}v^{9}}{8^{10}u^{7}v^{3}}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$8^{8-10}u^{7-7}v^{9-3}$$

Subtract 10 from 8.

$$8^{-2}u^{7-7}v^{9-3}$$

Subtract 7 from 7.

$$\frac{1}{64}u^{0}v^{9-3}$$

For any number a except 0, a^{0}=1.

$$\frac{1}{64}v^{9-3}$$

Subtract 3 from 9.

$$\frac{1}{64}u^{0}v^{6}$$

For any term t except 0, t^{0}=1.

$$\frac{1}{64}\times 1v^{6}$$

For any term t, t\times 1=t and 1t=t.

$$\frac{1}{64}v^{6}$$

Differentiate w.r.t. v

$$\frac{3v^{5}}{32}$$

Steps Using Definition of a Derivative

Cancel out 8^{8}v^{3}u^{7} in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}v}(\frac{v^{6}}{8^{2}})$$

Calculate 8 to the power of 2 and get 64.

$$\frac{\mathrm{d}}{\mathrm{d}v}(\frac{v^{6}}{64})$$

The derivative of ax^{n} is nax^{n-1}.

$$6\times \frac{1}{64}v^{6-1}$$

Multiply 6 times \frac{1}{64}.

$$\frac{3}{32}v^{6-1}$$

Subtract 1 from 6.

$$\frac{3}{32}v^{5}$$